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\title{多元统计分析练习3.3-3.5}
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\date{2024 年 3 月 26 日}
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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设 $x\sim N_p(\mu,\Sigma)$, 其中 $\Sigma$ 正定，
设 $x_1,x_2,\cdots,x_n$ 是从总体 $x$ 中抽取的一个简单随机样本。
求 $\mu$ 和 $\Sigma$ 的极大似然估计。

\vspace{0.2cm}

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\item  %Problem 02
验证 $\bar{x}$ 是 $\mu$ 的无偏估计，但 
$$\hat{\Sigma} = \frac{1}{n} \sum\limits_{i=1}^{n} (x_i-\bar{x})(x_i-\bar{x})' $$
不是 $\Sigma$ 的无偏估计。

\vspace{0.2cm}

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\item  %Problem 03
设有随机变量 $y$ 和随机向量 $x=(x_1,\cdots,x_p)'$, 设 
$$
E\begin{pmatrix} y \\ x \end{pmatrix} = \begin{pmatrix} \mu_y \\ \mu_x \end{pmatrix}, \,\,
V\begin{pmatrix} y \\ x \end{pmatrix} = \begin{pmatrix} \sigma_{yy} & \sigma_{xy}' \\ \sigma_{xy} & \Sigma_{xx}  \end{pmatrix}. 
$$
求 $y$ 和线性函数 $\ell' x$ 相关系数。其中 $\ell$ 是 $p$ 维常数向量。当 $\ell$ 任取时，求相关系数的最大值。

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\item  %Problem 04
证明随机变量 $x_1,\cdots,x_p$ 的任意线性函数 $F=a_1x_1+\cdots+a_px_p$ 与 $x=(x_1,\cdots,x_p)'$ 的复相关系数为1. 

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\item  %Problem 05
对31人进行人体测试，测试的7个指标是年龄、体重、肺活量、1.5英里跑的时间、休息时的脉搏、跑步时的脉搏、和跑步时的最大脉搏。求肺活量与其余六个指标的样本复相关系数。

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\item  %Problem 06
用随机向量 $x$ 的函数 $g(x)$ 来预测随机变量 $y$ 时，可用均方误差 $E[y-g(x)]^2$ 
作为预测精度的度量。如果限制 $g(x)$ 为线性函数，则使得均方误差达到最小的线性预测函数是什么？ 
最优线性预测的均方误差是什么？

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\item  %Problem 07
对31人进行人体测试，测试的7个指标是年龄，体重，肺活量，1.5英里跑的时间，休息时的脉搏，跑步时的脉搏，跑步时的最大脉搏。建立肺活量对其余六个变量的线性回归模型。计算复判定系数。计算肺活量与其余变量的复相关系数。

\vspace{0.2cm}

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\item  %Problem 08
当随机向量 $x_2$ 是偏变量时，随机向量 $x_1$ 的偏协方差矩阵定义为预测误差的协方差矩阵 $V(e)$, 其中的预测模型是用 $x_2$ 对 $x_1$ 的最优线性预测。写出偏协方差矩阵和偏相关系数的计算公式。

\vspace{0.2cm}

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\item  %Problem 09
设对16个婴儿测量了体重、出生天数和舒张压。
求样本相关矩阵。
求在控制出生天数后，舒张压和出生体重的样本偏相关系数。
求在控制出生体重后，舒张压和出生天数的样本偏相关系数。

\vspace{0.2cm}

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\item  %Problem 10
设 $x\sim N_p(0,\Sigma)$, 其中 $\Sigma$ 正定。
设 $x_1,x_2,\cdots,x_n$ 是从总体 $x$ 中抽取的一个简单随机样本。设 $n\ge p$. 
证明样本均值 $\bar{x}$ 与样本协方差矩阵 $S$ 相互独立。它们分别服从什么分布？

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\end{enumerate}


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\end{document}

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